A Logic for Statutes.

• Author: Lawsky, Sarah B.

1. Introduction 61 II. Defeasible Reasoning and Default Logic 64 III. Using Default Logic To Formalize the Law: An Example 66 A. The Law: Section 163 66 B. Section 163 Formalized 69 IV. Why Default Logic? 72 A. The Accuracy of Default Logic 72 B. The Benefits of Default Logic 77 V. Conclusion 80 I. Introduction

   Common law reasoning is, without question, a puzzle. When students are taught to "think like lawyers" in their first year of law school, they are taught case-based, common law reasoning. Books on legal reasoning--and there are many--are devoted almost entirely to case-based reasoning. Is case-based reasoning reasoning from analogy? How should such reasoning be modeled? How can it be justified?

   Rule-based legal reasoning, as exemplified by statutory reasoning, is in contrast taken as simple in legal scholarship. Statutory interpretation--how to determine the meaning of words in a statute, the relevance of the lawmakers' intent, and so forth--is much discussed, but there is little treatment of the structure of statutory reasoning once the meaning of the words is established. For example, in a leading book on legal reasoning, the chapter entitled "Interpreting Statutes and Other Posited Rules" addresses only the problems of interpreting the lawmakers' intended meaning. (1) The actual reasoning underlying statutory analysis is disposed of in just two pages: statutory reasoning simply involves following rules. Statutory reasoning is difficult only to the extent that understanding a term in the statute is difficult, and the meaning of the term, they explain, will be determined by a court, which throws us right back into common law reasoning. The classic text An Introduction to Legal Reasoning (2) deals with statutory reasoning in a similarly cursory fashion: statutory reasoning is often considered deductive, the book explains, and, while this may not be true, it is a useful approach; any complications that arise come from "ambiguity in the words used." (3)

   This Essay examines the structure of rule-based reasoning after ambiguities are resolved and the meaning of the rule's terms established. For rule-based legal reasoning is not best understood as merely deductive. And while rule-based reasoning can be fruitfully modeled using formal logic, standard formal logic is not the best approach for modeling rule-based legal reasoning. Rather, this Essay argues, using the Internal Revenue Code and accompanying regulations, judicial decisions, and rulings as its primary example, that at least some rule-based legal reasoning is best characterized as defeasible reasoning--reasoning that may result in conclusions that can be defeated by subsequent information--and is best modeled using a nonstandard logic called default logic. Default logic, unlike standard logic, directly represents defeasible reasoning: default logic permits formal reasoning that results in conclusions that may later be defeated.

   Default logic is superior to standard logic for representing certain statutory reasoning because, as the Essay explains, default logic makes explicit otherwise implicit reasoning and decisions about rule priority, decisions that are required to follow statutes and other legal rules. Default logic also captures the structure of legal rules, which standard logic entirely fails to do, and it more accurately reflects rule-based legal reasoning as actually practiced by lawyers, judges, and legislative drafters.

   Moreover, because default logic permits a more accurate representation of rule-based legal reasoning, there are a number of theoretical and practical advantages to using default logic, as opposed to standard logic, to model such reasoning. First, using default logic to model rule-based legal reasoning highlights the importance of and permits theorizing about rule priority. Second, because default logic captures the actual structure of the law, default logic makes it easier to translate (legal) code into (computer) code, which is particularly important given the growing use of artificial intelligence in legal practice, whether for e-discovery or for searching for the structure (as opposed to substance) of arguments that have been particularly effective. And, finally, because default logic allows legislative drafters to formalize their actual practice, drafters who use default logic will find it simpler to detect errors and ambiguities in legislation.

   A range of literature argues that legal reasoning is best understood as defeasible reasoning. (4) Indeed, the word "defeasibility" is borrowed from the law. (5) Yet these sources generally (though not entirely) neglect the intrinsic defeasibility of rule-based legal reasoning. Hage, for example, argues that legal reasoning may be defeasible, but his reasons for defeasibility include only that the burden of proof or the process of discovery may introduce new information, and that extralegal considerations may include implied exceptions to the law. (6) Vernon Walker argues for the application of default logic to the law but limits his discussion to reasoning about evidence (fact-finding). (7) There are a few examples of defeasible rule-based reasoning in the literature. John Horty, for example, provides a fictional example of a conflict between a federal and a state statute to illustrate default reasoning. (8) This Essay takes a similar approach, but instead of using a fictional example, it draws from an actual statute and demonstrates defeasibility intrinsic to the statute itself.

2. Defeasible Reasoning and Default Logic

   Once deductive reasoning provides a conclusion, nothing within deductive reasoning can unseat that conclusion. Consider a very basic deductive argument: "If A, then B. We know that A. Therefore, B." Given A, no additional information can shake the reasoner from B. (Of course, changing the information one has can change the conclusion. "I thought that if A, then B. But I was wrong. So although I have A, I cannot conclude B.") Because conclusions arrived at through deductive reasoning cannot be defeated by additional information, such conclusions are indefeasible.

   Most everyday reasoning, in contrast, leads to defeasible conclusions, conclusions that might be defeated by additional information. (Defeasible reasoning is sometimes referred to as the logic of jumping to conclusions.) In the classic example, someone learns that Tweety is a bird and concludes that Tweety can fly. But this conclusion is defeasible, because additional information could cause the reasoner to change his mind. For example, if the reasoner learns that Tweety is a penguin, he will conclude that Tweety can't in fact fly.

   Because deductive logic is indefeasible--regardless of additional information, a conclusion, once reached, will not be rejected--the formalization of deductive logic ("standard logic") is monotonic: once a conclusion is accepted, it cannot be rejected. In contrast, formalized defeasible logic is nonmonotonic: additional information can cause the reasoner to reject an earlier conclusion.

   There are a variety of ways to formalize nonmonotonic reasoning. This Essay uses a variant of default logic. (9) Under this approach, the reasoner has a set of propositional formulas, W, which we can informally think of as a world of facts; default rules (each default rule [delta], and the set of default rules D); and a relationship between the default rules (

   For example, consider trying to determine whether a particular person--call him Henry--can read. If the only information you have is that Henry lives in the United States (UnitedStates), you should conclude that he can read (Read). (10) If you learn, however, that Henry is five years old, you should conclude he cannot read, as most children in the United States do not read before age six (Young for younger than age six). These two rules together give us our set A of default rules, rules that might be defeated by each other or by other rules. These rules don't apply with certainty, but in general they are good guides to reasoning.

   More formally, if we know that Henry lives in the United States and Henry is five years old, we have [DELTA]=(W, D,

   W = {UnitedStates, Young}

   D = {[[delta].sub.1], [[delta].sub.2]

   [[delta].sub.1]: UnitedStates [right arrow] Read

   [[delta].sub.1]: Young [right arrow] [logical not] Read

   The "lower" the rule, the stronger, so here, [[delta].sub.1] dominates [[delta].sub.2]. That is, if both might apply, [[delta].sub.1] "beats" [[delta].sub.2].

   Given W, we reason as follows. First, accept everything in our world--W--and everything that we can prove (using regular, monotonic, standard logic) from that world. The only thing we can prove from our world and nothing else is UnitedStates and Young--i.e., that Henry lives in the United States and is young.

   Second, take the most dominant rule--here, [[delta].sub.1] Young [right arrow] [logical not] Read, that is, if a person is young, then in general he cannot read. (Notice that this is the most specific rule as well.) Adding [[delta].sub.1] Young [right arrow] [logical not] Read to what we believe won't create any contradictions, as all we believe is United-States and Young. So we add 8 to the things we believe, along with everything else we can derive from W along with [[delta].sub.r]. We now believe that Henry can't read--[logical not] Read--because from Young and Young [right arrow] [logical not] Read we can derive [logical not] Read.

   We then move to the next-most-dominant...